Sydney: We experience the world in three dimensions — length, width and height. But what does it really mean when mathematicians or physicists refer to a fourth dimension? Is it simply a larger kind of space? Or is it the “space-time” concept popularised by Einstein’s theory of relativity?

Attempts to picture four dimensions often lead to sketches of a “four-dimensional cube”. Yet our brains interpret drawings on flat paper in two dimensions — or at best three — making it extremely difficult to truly visualise a fourth spatial direction.

Despite this limitation, mathematics allows us to understand higher dimensions without directly picturing them.

Understanding dimension

A dimension describes the number of independent directions available in a space.

A straight line is one-dimensional: movement is possible forward or backward, but these are opposite directions along the same axis. A thin thread or rope can be thought of as nearly one-dimensional because its thickness is negligible compared to its length.

A flat surface — such as a football pitch or the surface of a balloon — is two-dimensional. Here, movement is possible in two independent directions: forward-backward and side-to-side.

Our everyday world is three-dimensional. In addition to forward and sideways movement, we can also move vertically — up and down.

Four-dimensional space introduces one more independent direction. This is why space-time is often described as four-dimensional: it combines three spatial dimensions with time, where movement forward or backward in time forms an additional direction.

One way to conceptualise four dimensions is to imagine a fully immersive three-dimensional film in which each frame is three-dimensional, and you can also fast-forward or rewind through time.

Learning from cubes

Mathematicians often use analogies from lower dimensions to grasp higher-dimensional ideas.

A two-dimensional “cube” is simply a square. To represent a three-dimensional cube on paper, we draw two squares and connect corresponding corners.

Using the same principle, a four-dimensional cube can be represented by drawing two three-dimensional cubes and linking their corners. This process can continue for five or more dimensions — though drawing them neatly requires patience and space.

While such sketches help count vertices and edges, they do not enable us to genuinely “see” four dimensions. Our perception remains confined to two- or three-dimensional interpretations.

Why knots fail in four dimensions

Knots work in three dimensions because one-dimensional ropes can loop around and entangle one another. A properly tied knot holds because the rope’s strands block each other’s movement — something climbers and sailors rely upon.

In four dimensions, however, knots would immediately unravel.

To understand why, consider a simpler analogy. Imagine two-dimensional ants living on a flat surface divided by a line. For them, the line is an uncrossable barrier. They cannot conceive of stepping over it.

If their world suddenly became three-dimensional, they could simply move upward slightly and bypass the line altogether.

Now apply this reasoning to ropes in three dimensions. Two perpendicular ropes will block each other if pulled in opposite directions. But if a fourth spatial direction were available, one rope could shift slightly into that new direction, avoid the other entirely, and then return.

From our perspective, it would appear as though the ropes passed through each other like ghosts.

Thinking in terms of a film analogy: if each three-dimensional moment is a frame, one rope could shift into a slightly different “frame” in the fourth dimension, move past the other rope, and then shift back.

Knots in hgher dimensions

Does this mean knotting is impossible beyond three dimensions? For ropes, yes — any knot tied in a one-dimensional object would come undone in four-dimensional space.

However, higher-dimensional knotting is still possible. In four dimensions, two-dimensional surfaces — such as balloon skins, sheets or tubular surfaces — can be knotted.

Mathematicians have identified a general rule: take the dimension of the object, double it, and add one. This gives the highest dimension of space in which that object can form a stable knot.

Under this rule, a one-dimensional rope can be knotted in at most three dimensions. A two-dimensional surface can remain knotted in up to five dimensions.

The study of knotted surfaces in four-dimensional space remains an active and fascinating area of mathematical research, offering insight into structures that are difficult to visualise yet fundamental to understanding higher-dimensional geometry. 

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